Michael Baake (Bielefeld)Random noble means substitutions
While the structure and geometry of primitive substitution rules is rather well understood, this is less so for random substitutions. We revisit an old example due to Godreche and Luck (from 1987), the random Fibonacci substitution, and extend it to the class of random noble means substitutions. Each family leads to a hull with positive entropy, although every member of the hull turns out to be a Meyer set. There is a canonical invariant measure on the hull, and the Meyer property guarantees that the pure point part of the spectrum is non-trivial. (Joint work with Markus Moll).

Veronika Brazdova (University College London)Computational Model of Brain Atrophy
(joint with Jorge Manuel Cardoso, Marie Chupin, Sebastien Ourselin and Louis Lemieux)
Neurodegenerative diseases (e.g., Alzheimer's disease) and epilepsy are accompanied by brain atrophy. However, a definitive diagnosis of diseases such as AD can only be made post-mortem. It is therefore crucial to develop non-invasive procedures to diagnose, monitor, and possibly predict, disease progression.
The current simulations approaches are descriptive, lacking mechanistic realism. In contrast, we present a new modelling approach based on a cellular removal mechanism in a high-resolution volumetric array representation of the brain, with each voxel assigned a tissue and brain region label. Atrophy is simulated by an iterative (randomized) regionalised and layered process of voxel deletion and replacement.
Atrophy rates can be specified (estimated) for each brain region and tissue class.

Alexander Clark (Leicester)Topological methods in the study of generalised tilings
We will informally discuss how many of the ideas already developed in the study of tilings can carry over to a more general setting of point patterns in manifolds. In the process of doing this, we shall consider some topological constructions that have proven useful in this study that might also be useful in the more traditional setting.

Pierre-Philippe Dechant (Durham)Recent developments in affine symmetry principles for non-crystallographic systems
We explore the possibility that extended icosahedral systems occurring in nature may be describable by an extension of the icosahedral group by a non-compact operation. We derive such extensions in two ways, firstly by direct Kac-Moody-type extension of the (non-crystallographic) H3 root system [1], and secondly by projection of the affine (crystallographic) D6 root system [2]. We discuss applications to the structure of viruses as well as that of nested fullerene shells, so called carbon onions [3].
[1] Novel Kac-Moody-type affine extensions of non-crystallographic Coxeter groups. P-P Dechant, C Bœhm, R Twarock: J. Phys. A: Math. Theor. 45, 285202 (2012).
[2] Affine extensions of non-crystallographic Coxeter groups induced by projection. P-P Dechant, C Bœhm, R Twarock: Journal of Mathematical Physics 54, 093508 (2013).
[3] Viruses and Fullerenes – Symmetry as a Common Thread? P-P Dechant, J Wardman, T Keef, R Twarock: Acta Crystallographica A 70, 162-167 (2014).

Paul Duncan (Strathclyde)Modelling the statics and dynamics of charged colloids in solution
Colloidal solutions contain particles with sizes ranging from nanometers to microns which are suspended in a liquid (e.g., emulsions such as ink or milk) or a gas (e.g., aerosols such smoke or fog). These solutions are part of every day life and also have a vast array of industrial applications in areas as diverse as oil extraction and food processing. Often, the particles in these systems are highly charged, and, unfortunately, the current mathematical models that are used, such as Poisson-Boltzmann theory, are unable to even qualitatively describe their behaviour in many important situations. The break down of the conventional theories is due primarily to the presence of strong fluctuations and correlations in the charge density. In this work, we develop a new theoretical approach which can accurately describe the structure, thermodynamics, and dynamics of highly charged colloidal solutions.

Ian Ford (University College London)Entropy production due to non-stationary heat conduction
(joint with Zac Laker and Henry Charlesworth, UCL)
Heat flow is an irreversible process that is accompanied by entropy production. We have investigated the character of this entropy production for a simple system of a Brownian particle, held in a potential well and subjected to a heat bath with a temperature that varies in both time and space. Using a framework of stochastic thermodynamics we demonstrate that while negative fluctuations in entropy production are commonplace, the total entropy production, as well as two contributions to the total, are all rigorously positive when averaged over all possible realisations of the motion. We show that there is a residual contribution to the entropy production that does not satisfy such a specific second law. In passing, we note that this system is an example where a principle of maximum mean entropy production, under certain constraints, can be used to identify the stochastic dynamical behaviour.

Franz Gähler (Bielefeld)Primitive substitutions for higher-dimensional paper-folding structures
(joint with Johan Nilsson, Bielefeld)
Paper-folding sequences are one of the well-known examples of aperiodically ordered structures. Some are known to be generated by a primitive substitution, which allows to prove important properties like unique ergodicity or pure point spectrum of the associated dynamical systems.
Recently, also higher-dimensional analogues of paper-folding structures have been proposed [S.I. Ben-Abraham et al., Acta Cryst. (2013) A69, 123-130], constructed via a recursive procedure (not a substitution). We show here that, in any dimension, these structures can also be generated by a primitive substitution. This allows us to prove that they give rise to dynamical systems (via the translation action on the hull) which are uniquely ergodic and have pure point spectrum. Knowledge of a generating substitution also allows to compute topological and dynamical invariants, as well as the complexity of these higher-dimensional paper-folding structures.

Alan Haynes (York)Constructing cut and project sets which are close to lattices
A natural problem is to understand how close cut-and-project sets are to being periodic. In this talk we will explain a recent proof (joint with Henna Koivusalo) that, in any irrational cut-and-project setup, it is always possible to choose acceptance domains in a nontrivial way so that the resulting separated nets are bounded distance to lattices.

Christian Huck (Bielefeld)Dynamical properties of k-free lattice points
We revisit the visible points of a lattice in Euclidean n-space together with their generalisations, the kth-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit. This is joint work with Michael Baake.

Des Johnston (Heriot-Watt)ln(exp(L)) = L and finite size scaling for first order phase transitions
(with Marco Müller and Wolfhard Janke, Leipzig University)
First-order transitions are ubiquitous in nature. It is possible to go quite a long way in discussing the scaling laws for such first-order transitions using a simple heuristic two-phase model. We note that the standard inverse system volume scaling for finite size corrections at a first-order phase transition (i.e. 1/L^3 or an L^3 lattice in 3D) is transmuted to 1/L^2 scaling if there is an exponential low-temperature phase degeneracy and present a high precision study of the effect in a 3D plaquette Hamiltonian (the "gonihedric" Ising model) and its dual.

Zoltan Kadar (Leeds)Geometrically defined extensions of the Temperley-Lieb algebras
Statistical physics invites us to employ diagram algebras, where transfer matrices are the algebra elements, i.e., the rectangular diagrams with a collection of lines and dots in them and multiplication is concatenation. After an illustrative introduction of the vast territory of their application, I will introduce a new class thereof which serves as a geometric interpolation between two well-studied classes and brings up many new open questions in statistical physics, invariant theory and representation theory.

Jonathan Keelan (Open)Arterial growth from optimisation principles
Arterial trees are important physiological system supplying nutrients and oxygen to tissue. Using optimisation techniques, we create large scale models of the arterial trees of organs which we find to closely follow the morphological and geometrical properties of in vivo arterial systems.

Jeroen Lamb (Imperial College)Bifurcations of random dynamical systems
We discuss early developments of a bifurcation theory for dynamical systems with noise. The ideas will be illustrated in the context of a specific example of a pitchfork bifurcation with additive noise. This is joint work with Mark Callaway, Martin Rasmussen, Doan Thai Son (Imperial College) and Christian Rodrigues (MPI Leipzig).

Gregory Maloney (Newcastle)Searching for substitution rules with n-fold symmetry
(joint with Franz Gähler, Bielefeld, and Eugene Kwan, Harvard)
In 1996, Danzer and Nischke discovered, for each integer n greater than 3, a family of edge-to-edge substitution rules on the set of triangles with angles that are integer multiples of 180/n. Their goal was to study the substitution tilings that arise from these rules. For certain values of n, such as 5 and 9, these tilings contain n-pointed stars — that is, local regions that are symmetric under rotation by 360/n — but for other values of n this does not appear to be the case. However, using ad hoc search methods, other substitution rules have been found on a subset of the Danzer-Nischke tiles for n = 7, and these substitution rules do give rise to tilings with 7-fold symmetry. It is natural to ask whether it is possible to find substitution rules with n-fold symmetry for higher values of n. Unfortunately, the ad hoc methods used for n = 7 are impractical for higher values of n, and so a more systematic approach is required.
I will describe a computer program designed to search for substitution rules on subsets of the Danzer-Nischke tiles. I will show some of the substitutions that have been found with this program, and describe a surprising topological property that some of them exhibit.

Nicolò Musmeci (King's College)Dynamic analysis of clustering on financial market data
In this talk I will show the application of the DBHT method [1] to a set of 342 US stocks daily prices during the time period between 1997 and 2012.
The DBHT method is a novel approach to extract cluster structure and to detect hierarchical organization in complex data-sets, it is based on the study of the properties of Planar Maximally Filtered Graphs (PMFG) [2] [3], it is deterministic, requires no a-priori parameters and it does not need any expert supervision.
In the case of financial data, the method yields a clustering set of stocks and a hierarchical structure of correlations. I will discuss the dynamical evolution of these clusters and structures and show results about their persistence over time, together with analyses about their varying similarity with the ICB Industrial Sectors classification. Comparison with Linkage method will be discussed. These measures point out peculiar behaviours in coincidence with the 2007-08 financial crisis [4].
[1] Won-Min Song, T. Di Matteo, T. Aste, "Hierarchical information clustering by means of topologically embedded graphs", PLoS One 7(3) (2012) e31929.
[2] T. Aste, T. Di Matteo, S. T. Hyde, "Complex networks on hyperbolic surfaces”, Physica A 346 (2005) 20-26.
[3] F. Pozzi, T. Di Matteo and T. Aste, "Spread of risk across financial markets: better to invest in the peripheries", Scientific Reports 3 (2013) 1665.
[4] N. Musmeci, T. Di Matteo, T. Aste, working paper 2013.

Tiago Pereira (Imperial)Dynamics in heterogeneous networks
Recent experiments show that heterogeneous networks exhibit non-global synchronization, that is, massively connected nodes synchronise while less connected nodes do not. Striking examples are found in the brain.
Indeed, synchronisation between highly connected neurons, called hub neurons, coordinate and shape the development in hippocampal networks.
Moreover, synchronisation among hubs neurons can drive epileptic seizures. I will discuss a probabilistic approach towards the explanation of this phenomenon in the limit networks of large random networks.
This is joint work with J. Lamb and S. van Strien.

Andrea Pizzoferrato (Warwick)Statistical mechanics of bipartite spin systems
(joint work with A. Barra, A. Galluzzi, F. Guerra and D. Tantari)
Inspired by a continuously increasing interest in modeling and framing complex systems in a thermodynamic rationale, in this talk we will try to explain the structure and properties of spin systems made up of a weighted combination of two parties of variables embedded with all the possible interactions (self and reciprocal). We will discuss the explicit expression for the free energy together with the range of validity of the thermodynamic limit. Moreover, for certain values of the weights, a critical surface on the graph of the free energy appears, which forces the system to have one degenerate self consistent equation for both the magnetisations of the two parties. The analysis is focused on both ferromagnetic and disordered interactions (at least at the replica symmetric level) using well known analytical interpolation techniques.

David Saad (Aston)Polymers, traffic and communication - from statistical physics to the London tube
Optimizing paths on networks is crucial for many applications, from subway traffic to Internet communication. As global path optimization that takes account of all path-choices simultaneously is computationally hard, most existing routing algorithms optimise paths individually, thus providing sub-optimal solutions. This work includes two different aspects of routing. In the first [1] we employ the cavity approach to study analytically the routing of nodes on a graph of given topology to predefined network routers and devise the corresponding distributive optimisation algorithm. In the second [2] we employ the physics of interacting polymers and disordered systems (the replica method) to analyse macroscopic properties of generic path-optimisation problems between arbitrarily selected communicating pairs; we also derive a simple, principled, generic and distributive routing algorithm capable of considering simultaneously all individual path choices.
Two types of nonlinear interactions are considered with different objectives: 1) alleviate traffic congestion at both cyber and real space and provide better route planning; and 2) save resources by powering down non-essential and redundant routers/stations at minimal cost. This saves energy and man-power, and alleviates the need for investment in infrastructure. We show that routing becomes more difficult as the number of communicating nodes increases and exhibits interesting physical phenomena such as ergodicity breaking. The ground state of such systems reveals non-monotonic complex behaviours in average path-length and algorithmic convergence, depending on the network topology, and densities of communicating nodes and routers.
We demonstrate the efficacy of the new algorithm [2] by applying it to: (i) random graphs resembling Internet overlay networks; (ii) travel on the London underground network based on Oyster-card data; and (iii) the global airport network. Analytically derived macroscopic properties give rise to insightful new routing phenomena, including phase transitions and scaling laws, which facilitate better understanding of the appropriate operational regimes and their limitations that are difficult to obtain otherwise.
[1] C. H. Yeung, D. Saad, The Competition for Shortest Paths on Sparse Graphs, Phys. Rev. Lett., 108, 208701 (2012).
[2] C. H. Yeung, D. Saad and K. Y. M. Wong, From the Physics of Interacting Polymers to Optimizing Routes on the London Underground, Proceedings of the National Academy of Science, 110, 13717-13722 (2013).

Yuzuru Sato (Hokkaido/Imperial College)Random dynamical systems approaches to noise-induced phenomena
Noise-induced phenomena arise out of interaction between deterministic dynamics and stochastic noise. Stochastic resonance, noise-induced synchronization, and noise-induced chaos are typical examples in statistical and nonlinear physics. In this talk, I present recently discovered noise-induced phenomena and their applications to nonlinear time-series analysis for dynamical systems with a large degrees of freedom based on experimental data of rotating fluid.

Geoffrey Sewell (Queen Mary)Relativistic thermodynamics of moving bodies
I address the long standing question of temperature transformations under Lorentz boosts by considering the thermal transactions between a body moving with uniform velocity relative to a certain inertial frame and a thermometer, designed to measure its temperature, that is held at rest in that frame.
On this basis I establish that the temperature reading is model dependent and thus that there is no intrinsic law of temperature transformations under Lorentz boosts.
I take this to signify that the concept of temperature is limited to bodies in their rest frames.

Stefanie Thiem (Oxford)Modelling the magnetic structure of rare-earth quasicrystals
We study the structure of the RKKY interactions and the corresponding low-temperature behaviour of magnetic moments for quasiperiodic tilings. The alignment of magnetic moments in rare-earth quasicrystals remains a fundamental open problem despite the continuous effort since the discovery of quasicrystals. We compute the RKKY interactions between the localised magnetic moments by means of a continuous fraction expansion of the Green’s function of the conduction electrons. Thus, our approach takes the structure of the critical electronic wave functions into account. The results show the emergence of strongly coupled spin clusters while the inter-cluster coupling is significantly weaker. Monte Carlo simulations reveal with decreasing temperature first the freezing of spins within the clusters followed by the freezing of the clusters. Thus, the low-temperature phase behaves like a cluster spin glass which is in good agreement with previous experimental findings.

Robert Turner (Nottingham)Fluctuating observation time ensembles and an analogous Jarzynski equality in the thermodynamics of trajectories
The dynamics of stochastic systems, both classical and quantum, can be studied by analysing the statistical properties of dynamical trajectories. The properties of ensembles of such trajectories for long, but fixed, times are described by large-deviation (LD) rate functions. These LD functions play the role of dynamical free-energies: they are cumulant generating functions for time-integrated ob- servables, and their analytic structure encodes dynamical phase behaviour. This "thermodynamics of trajectories" approach is to trajectories and dynamics what the equilibrium ensemble method of statistical mechanics is to con gurations and statics. We discuss how, just like in the static case, there is a variety of alternative ensembles of trajectories, each defined by their global constraints, with that of trajectories of fixed total time being just one of these. We show that an ensemble of trajectories where some time-extensive quantity is constant (and large) but where total observation time fluctuates, is equivalent to the fixed-time ensemble, and the LD functions that describe one ensemble can be obtained from those that describe the other. We discuss how the equivalence between generalised ensembles can be exploited in path sampling schemes for efficiently generating rare dynamical trajectories. We further develop equivalent fluctuation relations for the "thermodynamics of trajectories," focusing on an analog to the Jarzynski equality, and demonstrate its use in extracting dynamical free-energies from simulation.

James Walton (Leicester)Poincaré duality for pattern-equivariant (co)homology
Two aims in studying the topology of tiling spaces are, firstly, to understand how one may interpret topological invariants of tiling spaces and, secondly, to find ways of actually computing them for specific examples. In the first direction, I will show how one may interpret the Čech cohomology groups of a tiling in a highly geometric way, via a Poincaré duality result, using so called “pattern-equivariant chains” on the tiling. These groups have an analogous definition to the well-known pattern-equivariant cohomology groups. I will present an efficient method for computing these groups for hierarchical tilings. When considering the rotation-invariant versions of these groups, one often finds extra torsion in the calculated invariants to the Čech groups. Above being mere artefacts of the calculations, I will show how one may incorporate these extra torsion groups into a spectral sequence converging to the cohomology of the Euclidean hull of a tiling.

Gary Willis (Imperial College)Real-space renomalisation transformations on finite 2-d Ising and Potts lattices
Using and extending upon Hasenbusch's method, y_t and y_h were calculated numerically by performing a Real-Space Renomalisation transformation from a 4x4 to a 2x2 lattice, on both square and triangular lattices. Given the small sizes of the lattices involved, it comes as quite a surprise that both fall within 7% of their known analytic values. For the 3-state Potts model, one can again, after extending the method determine y_t and y_h numerically but we also find a third, thus far unexplained critical exponent whose original and significance remain a mystery.

Emilio Zappa (York)Structural transitions of icosahedral quasicrystals
As explained in [1], structural transitions of quasicrystals can be characterised in a group theoretical framework . In particular we have analysed the subgroup structure of the hyperoctahedral group in six dimensions [2], as a prerequisite to determine all possible structural transitions of three-dimensional icosahedral quasicrystals. In this talk I will first introduce this group theoretical classification and will then show (based on examples) how these can be used to construct structural transitions of this type.
[1] G. Indelicato et al., Structural transformations in quasicrystals induced by higher dimensional lattice transitions, Proceedings of the Royal Society A (2012), 468, 1452-1471.
[2] E. Zappa et al., On the subgroup structure of the hyperoctahedral group in six dimensions, arXiv:1402.3136.